If I have a three digit number like 293 Is there a method to compress it into one digit. Can any three digit number be rewritten into a single digit with some formula and then back to its original form?
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If someone answers this correctly, I would understand the $n=3$ case of this problem. – Sameer Baheti Jul 25 '20 at 07:24
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Would you accept Chinese 'digits'? – Chrystomath Jul 25 '20 at 07:30
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3I'm not sure I understand. There are 900 different 3 digit numbers (100 to 999) and 10 different single-digit numbers (in base 10). How can you map 10 numbers to 900 bijectively? – Graviton Jul 25 '20 at 07:32
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Use base $1000$ – Claude Leibovici Jul 25 '20 at 08:23
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No, assuming by three digit number you mean $100\leq n\leq 999$, then there are $999-100+1=900$ three digit numbers. But there are only $10$ single digit numbers (the integers $0$ through $9$). Since sets with different cardinalities can not have a bijection between them, there is no invertible function which maps three digit numbers to single digit numbers.
Of course, if you do not need the map to be invertible, then there are any number of functions you can use. For example
$$f(x)=\left\lfloor\frac{x}{100}\right\rfloor$$
simply maps every three digit number to its first digit (i.e. $f(293)=2$).
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